I am not familiar with the x250 Ducato, as mine is an earlier 2006 x244.
I assume that you mean that your number plate lamp is only staying on for about 5 minutes, when it should stay on. Not that the lamp is coming on at random.
I suspect that you a looking in the wrong area, as what you describe is not typical relay behaviour.
The light will probably be controlled by the BCM, which will have a programmed range of current to be drawn by the lamp. If the current drawn is deemed incorrect the BCM will raise an error warning, and if the current is seen as too high, perhaps switch off as a protective measure.
Check that correct wattage bulbs are fitted, to eliminate the above which is a guess on my part.
Also try new bulbs, to eliminate the possibility of an obscure bulb fault.